FIG. 1 is a prior-art diagram illustrating a fluid-flow measurement technique according to the prior art. A first ultrasonic transducer 110 is located at a wall 115 of a pipe 120 upstream from a second ultrasonic transducer 125 located further downstream at a wall 130 of the pipe 120.
A first ultrasonic signal is emitted from the upstream transducer 110 and received at the downstream transducer 125 across a path 135A. A time-of-flight (TOF) T(1,2) between emission and reception of the first ultrasonic signal is measured. A second ultrasonic signal is emitted from the downstream transducer 125 and received at the upstream transducer 110 across a path 135B. A TOF T(2,1) between emission and reception of the second ultrasonic signal is measured. The paths 135A and 135B are of equal length L. Each of the paths 135A and 135B lies at an angle θ to a longitudinal axis of the pipe 120.
The velocity of the ultrasonic signal traveling at the angle θ downstream is boosted by the fluid flow velocity through the pipe 120, thus decreasing the TOF(1,2). Likewise, the velocity of the ultrasonic signal traveling at the angle θ upstream is impeded by the fluid flow velocity through the pipe 120, thus increasing the TOF(2,1).
Specifically, the velocity of the first ultrasonic signal traversing the path 135A is the sum of the velocity C of the ultrasonic energy traveling through a stationary fluid of the type traversing the pipe 120 and a velocity vector component v of the fluid velocity U along the path 135A. U is the total velocity of the fluid flowing parallel to the longitudinal axis of the pipe 120. That is, the total velocity of the first ultrasonic signal traversing the path 135A of length L is equal to C+v. The TOF T(1,2) is therefor: T(1,2)=(distance)/(velocity)=L/(C+v).
Likewise, the velocity of the second ultrasonic signal traversing the path 135B is the difference between the velocity C of the ultrasonic energy traveling through a stationary fluid of the type traversing the pipe 120 and the velocity vector component v of the fluid velocity U along the path 135B. That is, the total velocity of the second ultrasonic signal traversing the path 135B of length L is equal to C−v. The TOF T(2,1) is therefor: T(2,1)=(distance)/(velocity)=L/(C−v).
The velocity C of the ultrasonic energy traveling through a stationary fluid is a constant for the particular fluid flowing through the pipe 120. Therefore, the measured T(1,2) and T(2,1) provide the two equations, above, in the unknowns v and L. Solving the two equations for v:
  v  =            L      2        ⁡          [                                    T            ⁡                          (                              2                ,                1                            )                                -                      T            ⁡                          (                              1                ,                2                            )                                                            T            ⁡                          (                              1                ,                2                            )                                *                      T            ⁡                          (                              2                ,                1                            )                                          ]      
However, the TOF measurements account only for the vector component v along the measurement paths 135A and 135B of the fluid flow velocity U. The entire fluid flow velocity U is equal to v/cos θ. Thus:
  U  =            L              2        ⁢        cos        ⁢                                  ⁢        θ              ⁡          [                                    T            ⁡                          (                              2                ,                1                            )                                -                      T            ⁡                          (                              1                ,                2                            )                                                            T            ⁡                          (                              1                ,                2                            )                                *                      T            ⁡                          (                              2                ,                1                            )                                          ]      